My first cut at this post was a fairly traditional introduction to one parameter subgroups of Lie groups. Along the way, I made this illustration showing the complicated global versus simple local behavior of one parameter subgroups:
But then, in writing down examples, I realized that the readers are likely to be intimately familiar with the multiplication of complex numbers, and so I decided to concentrate on the multiplication of nonzero complex numbers. These form a nice (real) 2-dimensional Lie group with very clear one-parameter subgroups.
So I decided to rewrite the post using simple examples only. You can get the abstract Lie theory from wikipedia, but I had to include the above illustration cause it’s too pretty to put in the bit bucket.